Trigonometric FunctionsBy Daria EiteneerTopics Covered:Reminder: relationship between degrees and radiansThe unit circleDefinitions of trigonometric functions for a right triangleDefinitions of trigonometric functions for a unit circleExact values for trigonometric functions of most commonly used anglesTrigonometric functions of any angle θ' in terms of angle θ in quadrant ITrigonometric functions of negative anglesSome useful relationships among trigonometric functionsDouble angle formulasHalf angle formulasAngle addition formulasSum, difference and product of trigonometric functionsGraphs of trigonometric functionsInverse trigonometric functionsPrincipal values for inverse trigonometric functionsRelations between inverse trigonometric functionsGraphs of inverse trigonometric functionsUsing trigonometric functions: components of a vectorUsing trigonometric functions: phase shift of a waveDerivatives of trigonometric functionsNote: All figures, unless otherwise specified, have a permission to be copied, distributedand/or modified under the terms of the GNU Free Documentation License, Version 1.2 orlater.Reminder: Relationship Between Degrees and RadiansA radian is defined as an angle subtended at the center of a circle for which the arc length isθequal to the radius of that circle (see Fig.1).Fig.1. Definition of a radian.The circumference of the circle is equal to 2 R, where R is the radius of the circle. Consequently,π360°=2 radians. Thus,π1 radian=360°/2 π ≈ 57.296°1° = (2 /360) radians π 0.01745 radians≈The Unit CircleIn mathematics, a unit circle is defined as a circle with a radius of 1. Often, especially inapplications to trigonometry, the unit circle is centered at the origin (0,0) in the coordinate plane.The equation of the unit circle in the coordinate plane isx2 + y2 = 1.As mentioned above, the unit circle is taken to be 360°, or 2 radians. We can divide theπcoordinate plane, and therefore, the unit circle, into 4 quadrants. The first quadrant is defined interms of coordinates by x>0, y>0, or, in terms of angles, by 0°< <90°, or 0< < /2. The secondθ θ πquadrant is defined by x<0, y>0, or 90°< <180°, or /2< < . The third quadrant is defined byθ π θ πx<0, y<0, or 180°< <270°, or < <3θ π θ π/2. Finally, the fourth quadrant is defined by x>0, y<0, or270°< <360°, or 3 /2< <2 .θ π θ πTrigonometric FunctionsDefinitions of Trigonometric Functions For a Right TriangleA right triangle is a triangle with a right angle (90°) (See Fig.2).Fig.2. Right triangle.For every angle in the triangle, there is the side of the triangle adjacent to it (from here onθdenoted as “adj”), the side opposite of it (from here on denoted as “opp”), and the hypotenuse(from here on denoted as “hyp”), which is the longest side of the triangle located opposite of theright angle. For angle , the trigonometric functions are defined as follows:θsine of = sin =θ θopphypcosine of = cos =θ θadjhyptangent of = tan =θ θsinθcosθ=oppadjcotangent of = cot =θ θ1tanθ=cosθsinθ=adjoppsecant of = sec =θ θ1cosθ=hypadjcosecant of = csc =θ θ1sinθ=hypoppDefinitions of Trigonometric Functions For a Unit CircleIn the unit circle, one can define the trigonometric functions cosine and sine as follows. If (x,y) isa point on the unit cirlce, and if the ray from the origin (0,0) to that point (x,y) makes an angle θwith the positive x-axis, (such that the counterclockwise direction is considered positive), then,cos = x/1 = xθsin = y/1 = yθThen, each point (x,y) on the unit circle can be written as (cos , sin ). Combined with theθ θequation x2 + y2 = 1, the definitions above give the relationship sin2+cosθ2=1. In addition, otherθtrigonometric functions can be defined in terms of x and y:tan = sin /cos = y/xθ θ θcot = cos /sin = x/yθ θ θsec = 1/cos = 1/xθ θcsc = 1/θ sin = 1/yθFig.3 below shows a unit circle in the coordinate plane, together with some useful values of angleθ, and the points (x,y)=(cos , sin ), that are most commonly used (also see table in the followingθ θsection).Fig.3. Most commonly used angles and points of the unit circle.Note: For in quadrant I, θ sinθ>0, cos θ >0; for in quadrant II, sin >0, cos <0; for in quadrantθ θ θ θIII, sin <0, cos <0; and for in quadrant IV, sin θ θ θ θExact Values for Trigonometric Functions of Most Commonly Used Angles in degreesθ in radiansθ sinθ cosθ tanθ0 0 0 1 030π612323345π42222160π33212390π21 0 undefined180 π 0 -1 02703 π2-1 0 undefined360 2π 0 1 0Note: Exact values for other trigonometric functions (such as cotθ, secθ, and cscθ) as well astrigonometric functions of many other angles can be derived by using the following sections.Trigonometric Functions of Any Angle θ' in Terms of Angle θ in Quadrant Iθ' sinθ' cosθ' tanθ' θ' sinθ' cosθ' tanθ'90°+θπ/2+θcosθ-sinθ-cotθ90°-θπ/2-θcosθsinθcotθ180°+θπ+θ-sinθ -cosθ tanθ180°-θπ-θsinθ -cosθ -tanθ270°+θ3π/2+θ-cosθ sinθ -cotθ270°-θ3π/2-θ-cosθ -sinθ cotθk(360°)+θk(2π)+θk=integersinθ cosθ tanθk(360°)-θk(2π)-θk=integer-sinθ cosθ -tanθTrigonometric Functions of Negative Anglessin(- ) = -sinθ θcos(- ) = cosθ θtan(- ) = -tanθ θSome Useful Relationships Among Trigonometric Functionssin2 + cosθ2 = 1θsec2 – tanθ2 = 1θcsc2 – cotθ2 = 1θDouble Angle Formulassin2 = 2 sin cosθ θ θcos2 = cosθ2θ – sin2θ = 1-2 sin2θ = 2 cos2θ -1tan2θ = 2tanθ1−tan2θHalf Angle FormulasNote: in the formulas in this section, the “+” sign is used in the quadrants where the respectivetrigonometric function is positive for angle /2, and the “-” sign is used in the quadrants where theθrespective trigonometric function is negative for angle /2.θsinθ2= ±1−cosθ2cosθ2= ±1cosθ2tanθ2= ±1−cosθ1cosθ=sinθ1cosθ=1−cosθsinθAngle Addition FormulasNote: in this and the following section, letters A and B are used to denote the angles of interest,instead of the letter .θsin A±B=sinAcosB± cosAsinBcos A± B=cosAcosB∓sinAsinBtan A± B=tanA± tanB1∓tanAtanBcot(A±B)=cotAcotB∓1cotB± cotASum, Difference and Product of Trigonometric FunctionssinA + sinB =2sinAB2cosA−B2sinA – sinB =2sinA−B2cosAB2cosA + cosB =2cosAB2cosA−B2cosA – cosB =−2sinAB2sinA−B2sinA sinB